Precalculus Examples

$x^{2} e^{x} - 5 x e^{x} = 0$

Step 1

Factor $x e^{x}$ out of $x^{2} e^{x} - 5 x e^{x}$ .

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Step 1.1

Factor $x e^{x}$ out of $x^{2} e^{x}$ .

$x e^{x} (x) - 5 x e^{x} = 0$

Step 1.2

Factor $x e^{x}$ out of $- 5 x e^{x}$ .

$x e^{x} (x) + x e^{x} (- 5) = 0$

Step 1.3

Factor $x e^{x}$ out of $x e^{x} (x) + x e^{x} (- 5)$ .

$x e^{x} (x - 5) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$e^{x} = 0$

$x - 5 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 4.2

Solve $e^{x} = 0$ for $x$ .

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Step 4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 4.2.3

There is no solution for $e^{x} = 0$

No solution

Step 5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 6

The final solution is all the values that make $x e^{x} (x - 5) = 0$ true.

$x = 0, 5$

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